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I tried to search online but I can't find the right definition. Does the first square bracket represent the domain of $x$ and the second square bracket the domain of $y$? Therefore:

does $(x, y)$ $ \in [10,-10] \times [0,0] $ mean a line from $(-10,0)$ to $(10,0)$?

The context:

enter image description here

Taylor Rendon
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underdisplayname
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  • IMO $x \in [10,-10]$ and $y \in [0,0]$ because $(x,y)$ usually means a pair i.e. an element of the cartesian product – Mauro ALLEGRANZA Jul 27 '20 at 08:54
  • So that's same as my interpretation? – underdisplayname Jul 27 '20 at 08:56
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    Note that there is a big difference between the notations $x,y\in \ldots$ and $(x,y)\in\ldots$. In the context, they use the second notation, but you ask for the first. – NeitherNor Jul 27 '20 at 09:07
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    I also think that there is a typo in the context, i.e. that what they wanted to say is $(x,y)\in[10,-10]\times[0,5]$, i.e. they want you to integrate along the upper half of the ellipse (they use $[10,-10]$ instead of $[-10,10]$ to indicate that you should integrate counter-clockwise) – NeitherNor Jul 27 '20 at 09:12

1 Answers1

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Lets look at the defintion of a Cartesian product of two sets:

Let $A$, $B$ both be sets such that $a \in A$ and $b \in B$. Then the set $A \times B$ $:=${$(a,b) | a \in A, b \in B$} is the cartesian product of the the sets $A$ and $B$. We read this set as "$A$ cross $B$".

Notice in defintion: $a \in A$ in the first slot and $b \in B$ in the second slot of the ordered pairs $(a,b)$ $\in A \times B$. So, for the set you mentioned:

When the author states that $(x,y) \in [10,-10] \times [0,0]$ (which we will assume to be a typo since the interval $[0,0]$ seems a little out of context here; the author probably meant $[10,-10] \times [0,5]$ as the domain) they mean that $x$ can be any value equal to or between $10$ and $-10$ and $y = 0$.

Remark: Note that $[10,-10], [0,0] \subset \mathbb{R}$ and $[10,-10] \times [0,0] \subset \mathbb{R}^2$. For a more geometrical interpretation as to why the author chose $[10,-10] \times [0,0]$ as the domain, see NeitherNor's comment.

EDIT: After thinking more about this problem at a geometric level, $[10,-10] \times [0,0] \subset \mathbb{R}^2$ being the domain is certainly not a typo. As you can see in the graph of the ellipse $\frac{x^{2}}{100}+\frac{y^{2}}{25} = 1$, on $\mathbb{R}^2$, below - the domain is certainly the line $[10,-10] \times [0,0]$ - of course in this problem, we are only studying the upper half of the ellipse.

enter image description here

Taylor Rendon
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    The author of this question (my prof) did not choose the domain $[10,−10]×[0,5]$, but $[10,−10]×[0,0]$. – underdisplayname Jul 27 '20 at 20:53
  • Of course. Myself and other users in the comments believe this to be a typo as it seems a bit out of context for the problem for the domain to be $[10,-10] \times [0,0]$. However, if this is not a typo, my answer is still precise up to the main notions of the cartesian product of two sets - and what the author is trying to convey. @soobster – Taylor Rendon Jul 27 '20 at 20:56
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    Thanks. So to conclude my understanding, if it was $[10,−10]×[0,0]$, then it would be a line in $\mathbb{R}^2$ and if it was $[10,−10]×[0,5]$, then it's a 2D area of the size 100. Am I right? – underdisplayname Jul 27 '20 at 21:00
  • If the author means $[10,-10] \times [0,0]$ the domain would be a line in $\mathbb{R}^2$ through the origin including the end points $10$ and $-10$. The cardinality (size) of the set (region) $[10, -10] \times [0,5]$ is the product of the cardinality of the set $[10,-10]$ and $[0,5]$. – Taylor Rendon Jul 27 '20 at 21:06
  • Please do not hesitate to reach out regarding any other questions! @soobster – Taylor Rendon Jul 27 '20 at 21:12
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    I have talked to the prof and he replied that it was not a type-o. If you look at the wording of the question carefully, it states:

    "that lies along (x,y) ..."

    However, I still don't agree with the validity of the question. The ellipse does not "lie" along $$(x,y) \in [10,-10] \times [0,0]$$ which as we agreed, is a line.

     – underdisplayname Jul 27 '20 at 23:01
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    I think the wording should have been like: Find $\int F \cdot dr$ where $\gamma$ is the curve in the upper half of the x-y plane. – underdisplayname Jul 27 '20 at 23:04
  • @soobster - Please see the edit I made to the answer, it will make much more sense now! I agree with your second comment about the wording, by the way. – Taylor Rendon Jul 28 '20 at 01:30